# Relaxation of One-dimensional Collisionless Gravitating Systems

## TL;DR

This paper investigates the collisionless relaxation of one-dimensional gravitational systems, analyzing entropy behaviors and phase mixing effects through analytical and numerical methods, highlighting differences from collisional relaxation.

## Contribution

It introduces a Hermite-Legendre expansion approach to study entropy evolution and clarifies the roles of phase mixing versus violent relaxation in these systems.

## Key findings

- Fine-grained entropy remains time-independent.
- Coarse-grained entropy generally increases with oscillations.
- Phase mixing dominates over violent relaxation in entropy evolution.

## Abstract

In an effort to better understand collisionless relaxation processes in gravitational systems, we investigate one-dimensional models. Taking advantage of a Hermite-Legendre expansion of relevant distribution functions, we present analytical and numerical behaviors of Maxwell-Boltzmann entropy. In particular, we modestly perturb systems about a separable-solution equilibrium and observe their collisionless evolution to a steady state. We verify the time-independence of fine-grained entropy in these systems before turning our attention to the behavior of coarse-grained entropy. We also verify that there is no analogue to the collisional H-theorem for these systems. Competing terms in the second-order coarse-grained entropy make it impossible to guarantee continuously increasing entropy. However, over dynamical time-scales the coarse-grained entropy generally increases, with small oscillations occurring. The lack of substantive differences between the entropies in test-particle and self-gravitating cases suggests that phase mixing, rather than violent relaxation associated with potential changes, more significantly drives the coarse-grained entropy evolution. The effects of violent relaxation can be better quantified through analysis of energy distributions rather than phase-space distributions.